(C2xDic3).9D4 - GroupNames

G = (C₂×Dic₃).₉D₄ order 192 = 2⁶·3

3^rd non-split extension by C₂×Dic₃ of D₄ acting via D₄/C₂=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C₂×Dic₃).₉D₄, (C₂×Dic₃).₂Q₈, (C₂²×C₄).₂₇D₆, C₂².₄₀(S₃×Q₈), C₂.₉(D₆⋊Q₈), C₂².₁₅₂(S₃×D₄), C₆.₂₂(C₂²⋊Q₈), C₆.₁(C₄²⋊₂C₂), C₂.₄(C₄²⋊₃S₃), C₂.₉(C₂³.₉D₆), C₆.₁₆(C₄.₄D₄), C₂.₉(Dic₃.Q₈), C₂.C₄².₅S₃, C₆.C₄².₆C₂, (C₂²×C₁₂).₈C₂², C₆.₁₂(C₄².C₂), C₂².₈₅(C₄○D₁₂), (C₂²×C₆).₂₈₃C₂³, C₂³.₃₆₄(C₂²×S₃), C₆.₅(C₂².D₄), C₂.₈(C₂³.₈D₆), C₂².₈₃(D₄⋊₂S₃), C₂.₈(C₂³.₁₁D₆), C₃⋊₁(C₂³.₈₃C₂³), (C₂²×Dic₃).₈C₂², (C₂×C₆).₆₄(C₂×Q₈), (C₂×C₆).₁₉₅(C₂×D₄), (C₂×Dic₃⋊C₄).₇C₂, (C₂×C₆).₁₂₇(C₄○D₄), (C₃×C₂.C₄²).₁C₂, SmallGroup(192,217)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂²×C₆ — (C₂×Dic₃).₉D₄

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₂²×C₆ — C₂²×Dic₃ — C₆.C₄² — (C₂×Dic₃).₉D₄

Lower central

C₃ — C₂²×C₆ — (C₂×Dic₃).₉D₄

Upper central

C₁ — C₂³ — C₂.C₄²

Generators and relations for (C₂×Dic₃).₉D₄
G = < a,b,c,d,e | a²=b⁶=d⁴=1, c²=b³, e²=ab³, ab=ba, dcd^-1=ac=ca, ad=da, ae=ea, cbc^-1=dbd^-1=ebe^-1=b^-1, ece^-1=ab³c, ede^-1=b³d^-1 >

Subgroups: 336 in 134 conjugacy classes, 55 normal (51 characteristic)
C₁, C₂, C₃, C₄, C₂², C₆, C₂×C₄, C₂³, Dic₃, C₁₂, C₂×C₆, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×Dic₃, C₂×Dic₃, C₂×C₁₂, C₂²×C₆, C₂.C₄², C₂.C₄², C₂×C₄⋊C₄, Dic₃⋊C₄, C₂²×Dic₃, C₂²×C₁₂, C₂³.₈₃C₂³, C₆.C₄², C₃×C₂.C₄², C₂×Dic₃⋊C₄, (C₂×Dic₃).₉D₄
Quotients: C₁, C₂, C₂², S₃, D₄, Q₈, C₂³, D₆, C₂×D₄, C₂×Q₈, C₄○D₄, C₂²×S₃, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₄².C₂, C₄²⋊₂C₂, C₄○D₁₂, S₃×D₄, D₄⋊₂S₃, S₃×Q₈, C₂³.₈₃C₂³, C₄²⋊₃S₃, C₂³.₈D₆, C₂³.₉D₆, C₂³.₁₁D₆, Dic₃.Q₈, D₆⋊Q₈, (C₂×Dic₃).₉D₄

Smallest permutation representation of (C₂×Dic₃).₉D₄
►Regular action on 192 points

Generators in S₁₉₂

(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 34)(8 35)(9 36)(10 31)(11 32)(12 33)(19 188)(20 189)(21 190)(22 191)(23 192)(24 187)(25 42)(26 37)(27 38)(28 39)(29 40)(30 41)(43 56)(44 57)(45 58)(46 59)(47 60)(48 55)(49 66)(50 61)(51 62)(52 63)(53 64)(54 65)(67 80)(68 81)(69 82)(70 83)(71 84)(72 79)(73 90)(74 85)(75 86)(76 87)(77 88)(78 89)(91 105)(92 106)(93 107)(94 108)(95 103)(96 104)(97 113)(98 114)(99 109)(100 110)(101 111)(102 112)(115 129)(116 130)(117 131)(118 132)(119 127)(120 128)(121 137)(122 138)(123 133)(124 134)(125 135)(126 136)(139 153)(140 154)(141 155)(142 156)(143 151)(144 152)(145 161)(146 162)(147 157)(148 158)(149 159)(150 160)(163 177)(164 178)(165 179)(166 180)(167 175)(168 176)(169 185)(170 186)(171 181)(172 182)(173 183)(174 184)

(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)(25 26 27 28 29 30)(31 32 33 34 35 36)(37 38 39 40 41 42)(43 44 45 46 47 48)(49 50 51 52 53 54)(55 56 57 58 59 60)(61 62 63 64 65 66)(67 68 69 70 71 72)(73 74 75 76 77 78)(79 80 81 82 83 84)(85 86 87 88 89 90)(91 92 93 94 95 96)(97 98 99 100 101 102)(103 104 105 106 107 108)(109 110 111 112 113 114)(115 116 117 118 119 120)(121 122 123 124 125 126)(127 128 129 130 131 132)(133 134 135 136 137 138)(139 140 141 142 143 144)(145 146 147 148 149 150)(151 152 153 154 155 156)(157 158 159 160 161 162)(163 164 165 166 167 168)(169 170 171 172 173 174)(175 176 177 178 179 180)(181 182 183 184 185 186)(187 188 189 190 191 192)

(1 161 4 158)(2 160 5 157)(3 159 6 162)(7 168 10 165)(8 167 11 164)(9 166 12 163)(13 145 16 148)(14 150 17 147)(15 149 18 146)(19 37 22 40)(20 42 23 39)(21 41 24 38)(25 192 28 189)(26 191 29 188)(27 190 30 187)(31 179 34 176)(32 178 35 175)(33 177 36 180)(43 169 46 172)(44 174 47 171)(45 173 48 170)(49 114 52 111)(50 113 53 110)(51 112 54 109)(55 186 58 183)(56 185 59 182)(57 184 60 181)(61 97 64 100)(62 102 65 99)(63 101 66 98)(67 127 70 130)(68 132 71 129)(69 131 72 128)(73 138 76 135)(74 137 77 134)(75 136 78 133)(79 120 82 117)(80 119 83 116)(81 118 84 115)(85 121 88 124)(86 126 89 123)(87 125 90 122)(91 151 94 154)(92 156 95 153)(93 155 96 152)(103 139 106 142)(104 144 107 141)(105 143 108 140)

(1 110 32 127)(2 109 33 132)(3 114 34 131)(4 113 35 130)(5 112 36 129)(6 111 31 128)(7 117 15 98)(8 116 16 97)(9 115 17 102)(10 120 18 101)(11 119 13 100)(12 118 14 99)(19 108 185 85)(20 107 186 90)(21 106 181 89)(22 105 182 88)(23 104 183 87)(24 103 184 86)(25 141 45 122)(26 140 46 121)(27 139 47 126)(28 144 48 125)(29 143 43 124)(30 142 44 123)(37 154 59 137)(38 153 60 136)(39 152 55 135)(40 151 56 134)(41 156 57 133)(42 155 58 138)(49 165 69 146)(50 164 70 145)(51 163 71 150)(52 168 72 149)(53 167 67 148)(54 166 68 147)(61 178 83 161)(62 177 84 160)(63 176 79 159)(64 175 80 158)(65 180 81 157)(66 179 82 162)(73 189 93 170)(74 188 94 169)(75 187 95 174)(76 192 96 173)(77 191 91 172)(78 190 92 171)

(1 140 16 151)(2 139 17 156)(3 144 18 155)(4 143 13 154)(5 142 14 153)(6 141 15 152)(7 135 31 122)(8 134 32 121)(9 133 33 126)(10 138 34 125)(11 137 35 124)(12 136 36 123)(19 61 191 53)(20 66 192 52)(21 65 187 51)(22 64 188 50)(23 63 189 49)(24 62 190 54)(25 101 39 114)(26 100 40 113)(27 99 41 112)(28 98 42 111)(29 97 37 110)(30 102 38 109)(43 116 59 127)(44 115 60 132)(45 120 55 131)(46 119 56 130)(47 118 57 129)(48 117 58 128)(67 185 83 172)(68 184 84 171)(69 183 79 170)(70 182 80 169)(71 181 81 174)(72 186 82 173)(73 168 87 179)(74 167 88 178)(75 166 89 177)(76 165 90 176)(77 164 85 175)(78 163 86 180)(91 145 108 158)(92 150 103 157)(93 149 104 162)(94 148 105 161)(95 147 106 160)(96 146 107 159)

G:=sub<Sym(192)| (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,34)(8,35)(9,36)(10,31)(11,32)(12,33)(19,188)(20,189)(21,190)(22,191)(23,192)(24,187)(25,42)(26,37)(27,38)(28,39)(29,40)(30,41)(43,56)(44,57)(45,58)(46,59)(47,60)(48,55)(49,66)(50,61)(51,62)(52,63)(53,64)(54,65)(67,80)(68,81)(69,82)(70,83)(71,84)(72,79)(73,90)(74,85)(75,86)(76,87)(77,88)(78,89)(91,105)(92,106)(93,107)(94,108)(95,103)(96,104)(97,113)(98,114)(99,109)(100,110)(101,111)(102,112)(115,129)(116,130)(117,131)(118,132)(119,127)(120,128)(121,137)(122,138)(123,133)(124,134)(125,135)(126,136)(139,153)(140,154)(141,155)(142,156)(143,151)(144,152)(145,161)(146,162)(147,157)(148,158)(149,159)(150,160)(163,177)(164,178)(165,179)(166,180)(167,175)(168,176)(169,185)(170,186)(171,181)(172,182)(173,183)(174,184), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96)(97,98,99,100,101,102)(103,104,105,106,107,108)(109,110,111,112,113,114)(115,116,117,118,119,120)(121,122,123,124,125,126)(127,128,129,130,131,132)(133,134,135,136,137,138)(139,140,141,142,143,144)(145,146,147,148,149,150)(151,152,153,154,155,156)(157,158,159,160,161,162)(163,164,165,166,167,168)(169,170,171,172,173,174)(175,176,177,178,179,180)(181,182,183,184,185,186)(187,188,189,190,191,192), (1,161,4,158)(2,160,5,157)(3,159,6,162)(7,168,10,165)(8,167,11,164)(9,166,12,163)(13,145,16,148)(14,150,17,147)(15,149,18,146)(19,37,22,40)(20,42,23,39)(21,41,24,38)(25,192,28,189)(26,191,29,188)(27,190,30,187)(31,179,34,176)(32,178,35,175)(33,177,36,180)(43,169,46,172)(44,174,47,171)(45,173,48,170)(49,114,52,111)(50,113,53,110)(51,112,54,109)(55,186,58,183)(56,185,59,182)(57,184,60,181)(61,97,64,100)(62,102,65,99)(63,101,66,98)(67,127,70,130)(68,132,71,129)(69,131,72,128)(73,138,76,135)(74,137,77,134)(75,136,78,133)(79,120,82,117)(80,119,83,116)(81,118,84,115)(85,121,88,124)(86,126,89,123)(87,125,90,122)(91,151,94,154)(92,156,95,153)(93,155,96,152)(103,139,106,142)(104,144,107,141)(105,143,108,140), (1,110,32,127)(2,109,33,132)(3,114,34,131)(4,113,35,130)(5,112,36,129)(6,111,31,128)(7,117,15,98)(8,116,16,97)(9,115,17,102)(10,120,18,101)(11,119,13,100)(12,118,14,99)(19,108,185,85)(20,107,186,90)(21,106,181,89)(22,105,182,88)(23,104,183,87)(24,103,184,86)(25,141,45,122)(26,140,46,121)(27,139,47,126)(28,144,48,125)(29,143,43,124)(30,142,44,123)(37,154,59,137)(38,153,60,136)(39,152,55,135)(40,151,56,134)(41,156,57,133)(42,155,58,138)(49,165,69,146)(50,164,70,145)(51,163,71,150)(52,168,72,149)(53,167,67,148)(54,166,68,147)(61,178,83,161)(62,177,84,160)(63,176,79,159)(64,175,80,158)(65,180,81,157)(66,179,82,162)(73,189,93,170)(74,188,94,169)(75,187,95,174)(76,192,96,173)(77,191,91,172)(78,190,92,171), (1,140,16,151)(2,139,17,156)(3,144,18,155)(4,143,13,154)(5,142,14,153)(6,141,15,152)(7,135,31,122)(8,134,32,121)(9,133,33,126)(10,138,34,125)(11,137,35,124)(12,136,36,123)(19,61,191,53)(20,66,192,52)(21,65,187,51)(22,64,188,50)(23,63,189,49)(24,62,190,54)(25,101,39,114)(26,100,40,113)(27,99,41,112)(28,98,42,111)(29,97,37,110)(30,102,38,109)(43,116,59,127)(44,115,60,132)(45,120,55,131)(46,119,56,130)(47,118,57,129)(48,117,58,128)(67,185,83,172)(68,184,84,171)(69,183,79,170)(70,182,80,169)(71,181,81,174)(72,186,82,173)(73,168,87,179)(74,167,88,178)(75,166,89,177)(76,165,90,176)(77,164,85,175)(78,163,86,180)(91,145,108,158)(92,150,103,157)(93,149,104,162)(94,148,105,161)(95,147,106,160)(96,146,107,159)>;

G:=Group( (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,34)(8,35)(9,36)(10,31)(11,32)(12,33)(19,188)(20,189)(21,190)(22,191)(23,192)(24,187)(25,42)(26,37)(27,38)(28,39)(29,40)(30,41)(43,56)(44,57)(45,58)(46,59)(47,60)(48,55)(49,66)(50,61)(51,62)(52,63)(53,64)(54,65)(67,80)(68,81)(69,82)(70,83)(71,84)(72,79)(73,90)(74,85)(75,86)(76,87)(77,88)(78,89)(91,105)(92,106)(93,107)(94,108)(95,103)(96,104)(97,113)(98,114)(99,109)(100,110)(101,111)(102,112)(115,129)(116,130)(117,131)(118,132)(119,127)(120,128)(121,137)(122,138)(123,133)(124,134)(125,135)(126,136)(139,153)(140,154)(141,155)(142,156)(143,151)(144,152)(145,161)(146,162)(147,157)(148,158)(149,159)(150,160)(163,177)(164,178)(165,179)(166,180)(167,175)(168,176)(169,185)(170,186)(171,181)(172,182)(173,183)(174,184), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96)(97,98,99,100,101,102)(103,104,105,106,107,108)(109,110,111,112,113,114)(115,116,117,118,119,120)(121,122,123,124,125,126)(127,128,129,130,131,132)(133,134,135,136,137,138)(139,140,141,142,143,144)(145,146,147,148,149,150)(151,152,153,154,155,156)(157,158,159,160,161,162)(163,164,165,166,167,168)(169,170,171,172,173,174)(175,176,177,178,179,180)(181,182,183,184,185,186)(187,188,189,190,191,192), (1,161,4,158)(2,160,5,157)(3,159,6,162)(7,168,10,165)(8,167,11,164)(9,166,12,163)(13,145,16,148)(14,150,17,147)(15,149,18,146)(19,37,22,40)(20,42,23,39)(21,41,24,38)(25,192,28,189)(26,191,29,188)(27,190,30,187)(31,179,34,176)(32,178,35,175)(33,177,36,180)(43,169,46,172)(44,174,47,171)(45,173,48,170)(49,114,52,111)(50,113,53,110)(51,112,54,109)(55,186,58,183)(56,185,59,182)(57,184,60,181)(61,97,64,100)(62,102,65,99)(63,101,66,98)(67,127,70,130)(68,132,71,129)(69,131,72,128)(73,138,76,135)(74,137,77,134)(75,136,78,133)(79,120,82,117)(80,119,83,116)(81,118,84,115)(85,121,88,124)(86,126,89,123)(87,125,90,122)(91,151,94,154)(92,156,95,153)(93,155,96,152)(103,139,106,142)(104,144,107,141)(105,143,108,140), (1,110,32,127)(2,109,33,132)(3,114,34,131)(4,113,35,130)(5,112,36,129)(6,111,31,128)(7,117,15,98)(8,116,16,97)(9,115,17,102)(10,120,18,101)(11,119,13,100)(12,118,14,99)(19,108,185,85)(20,107,186,90)(21,106,181,89)(22,105,182,88)(23,104,183,87)(24,103,184,86)(25,141,45,122)(26,140,46,121)(27,139,47,126)(28,144,48,125)(29,143,43,124)(30,142,44,123)(37,154,59,137)(38,153,60,136)(39,152,55,135)(40,151,56,134)(41,156,57,133)(42,155,58,138)(49,165,69,146)(50,164,70,145)(51,163,71,150)(52,168,72,149)(53,167,67,148)(54,166,68,147)(61,178,83,161)(62,177,84,160)(63,176,79,159)(64,175,80,158)(65,180,81,157)(66,179,82,162)(73,189,93,170)(74,188,94,169)(75,187,95,174)(76,192,96,173)(77,191,91,172)(78,190,92,171), (1,140,16,151)(2,139,17,156)(3,144,18,155)(4,143,13,154)(5,142,14,153)(6,141,15,152)(7,135,31,122)(8,134,32,121)(9,133,33,126)(10,138,34,125)(11,137,35,124)(12,136,36,123)(19,61,191,53)(20,66,192,52)(21,65,187,51)(22,64,188,50)(23,63,189,49)(24,62,190,54)(25,101,39,114)(26,100,40,113)(27,99,41,112)(28,98,42,111)(29,97,37,110)(30,102,38,109)(43,116,59,127)(44,115,60,132)(45,120,55,131)(46,119,56,130)(47,118,57,129)(48,117,58,128)(67,185,83,172)(68,184,84,171)(69,183,79,170)(70,182,80,169)(71,181,81,174)(72,186,82,173)(73,168,87,179)(74,167,88,178)(75,166,89,177)(76,165,90,176)(77,164,85,175)(78,163,86,180)(91,145,108,158)(92,150,103,157)(93,149,104,162)(94,148,105,161)(95,147,106,160)(96,146,107,159) );

G=PermutationGroup([[(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,34),(8,35),(9,36),(10,31),(11,32),(12,33),(19,188),(20,189),(21,190),(22,191),(23,192),(24,187),(25,42),(26,37),(27,38),(28,39),(29,40),(30,41),(43,56),(44,57),(45,58),(46,59),(47,60),(48,55),(49,66),(50,61),(51,62),(52,63),(53,64),(54,65),(67,80),(68,81),(69,82),(70,83),(71,84),(72,79),(73,90),(74,85),(75,86),(76,87),(77,88),(78,89),(91,105),(92,106),(93,107),(94,108),(95,103),(96,104),(97,113),(98,114),(99,109),(100,110),(101,111),(102,112),(115,129),(116,130),(117,131),(118,132),(119,127),(120,128),(121,137),(122,138),(123,133),(124,134),(125,135),(126,136),(139,153),(140,154),(141,155),(142,156),(143,151),(144,152),(145,161),(146,162),(147,157),(148,158),(149,159),(150,160),(163,177),(164,178),(165,179),(166,180),(167,175),(168,176),(169,185),(170,186),(171,181),(172,182),(173,183),(174,184)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24),(25,26,27,28,29,30),(31,32,33,34,35,36),(37,38,39,40,41,42),(43,44,45,46,47,48),(49,50,51,52,53,54),(55,56,57,58,59,60),(61,62,63,64,65,66),(67,68,69,70,71,72),(73,74,75,76,77,78),(79,80,81,82,83,84),(85,86,87,88,89,90),(91,92,93,94,95,96),(97,98,99,100,101,102),(103,104,105,106,107,108),(109,110,111,112,113,114),(115,116,117,118,119,120),(121,122,123,124,125,126),(127,128,129,130,131,132),(133,134,135,136,137,138),(139,140,141,142,143,144),(145,146,147,148,149,150),(151,152,153,154,155,156),(157,158,159,160,161,162),(163,164,165,166,167,168),(169,170,171,172,173,174),(175,176,177,178,179,180),(181,182,183,184,185,186),(187,188,189,190,191,192)], [(1,161,4,158),(2,160,5,157),(3,159,6,162),(7,168,10,165),(8,167,11,164),(9,166,12,163),(13,145,16,148),(14,150,17,147),(15,149,18,146),(19,37,22,40),(20,42,23,39),(21,41,24,38),(25,192,28,189),(26,191,29,188),(27,190,30,187),(31,179,34,176),(32,178,35,175),(33,177,36,180),(43,169,46,172),(44,174,47,171),(45,173,48,170),(49,114,52,111),(50,113,53,110),(51,112,54,109),(55,186,58,183),(56,185,59,182),(57,184,60,181),(61,97,64,100),(62,102,65,99),(63,101,66,98),(67,127,70,130),(68,132,71,129),(69,131,72,128),(73,138,76,135),(74,137,77,134),(75,136,78,133),(79,120,82,117),(80,119,83,116),(81,118,84,115),(85,121,88,124),(86,126,89,123),(87,125,90,122),(91,151,94,154),(92,156,95,153),(93,155,96,152),(103,139,106,142),(104,144,107,141),(105,143,108,140)], [(1,110,32,127),(2,109,33,132),(3,114,34,131),(4,113,35,130),(5,112,36,129),(6,111,31,128),(7,117,15,98),(8,116,16,97),(9,115,17,102),(10,120,18,101),(11,119,13,100),(12,118,14,99),(19,108,185,85),(20,107,186,90),(21,106,181,89),(22,105,182,88),(23,104,183,87),(24,103,184,86),(25,141,45,122),(26,140,46,121),(27,139,47,126),(28,144,48,125),(29,143,43,124),(30,142,44,123),(37,154,59,137),(38,153,60,136),(39,152,55,135),(40,151,56,134),(41,156,57,133),(42,155,58,138),(49,165,69,146),(50,164,70,145),(51,163,71,150),(52,168,72,149),(53,167,67,148),(54,166,68,147),(61,178,83,161),(62,177,84,160),(63,176,79,159),(64,175,80,158),(65,180,81,157),(66,179,82,162),(73,189,93,170),(74,188,94,169),(75,187,95,174),(76,192,96,173),(77,191,91,172),(78,190,92,171)], [(1,140,16,151),(2,139,17,156),(3,144,18,155),(4,143,13,154),(5,142,14,153),(6,141,15,152),(7,135,31,122),(8,134,32,121),(9,133,33,126),(10,138,34,125),(11,137,35,124),(12,136,36,123),(19,61,191,53),(20,66,192,52),(21,65,187,51),(22,64,188,50),(23,63,189,49),(24,62,190,54),(25,101,39,114),(26,100,40,113),(27,99,41,112),(28,98,42,111),(29,97,37,110),(30,102,38,109),(43,116,59,127),(44,115,60,132),(45,120,55,131),(46,119,56,130),(47,118,57,129),(48,117,58,128),(67,185,83,172),(68,184,84,171),(69,183,79,170),(70,182,80,169),(71,181,81,174),(72,186,82,173),(73,168,87,179),(74,167,88,178),(75,166,89,177),(76,165,90,176),(77,164,85,175),(78,163,86,180),(91,145,108,158),(92,150,103,157),(93,149,104,162),(94,148,105,161),(95,147,106,160),(96,146,107,159)]])

42 conjugacy classes

class	1	2A	···	2G	3	4A	···	4F	4G	···	4N	6A	···	6G	12A	···	12L
order	1	2	···	2	3	4	···	4	4	···	4	6	···	6	12	···	12
size	1	1	···	1	2	4	···	4	12	···	12	2	···	2	4	···	4

42 irreducible representations

dim	1	1	1	1	2	2	2	2	2	2	4	4	4
type	+	+	+	+	+	+	-	+			+	-	-
image	C₁	C₂	C₂	C₂	S₃	D₄	Q₈	D₆	C₄○D₄	C₄○D₁₂	S₃×D₄	D₄⋊₂S₃	S₃×Q₈
kernel	(C₂×Dic₃).₉D₄	C₆.C₄²	C₃×C₂.C₄²	C₂×Dic₃⋊C₄	C₂.C₄²	C₂×Dic₃	C₂×Dic₃	C₂²×C₄	C₂×C₆	C₂²	C₂²	C₂²	C₂²
# reps	1	4	1	2	1	2	2	3	10	12	1	2	1

Matrix representation of (C₂×Dic₃).₉D₄ ►in GL₆(𝔽₁₃)

12	0	0	0	0	0
0	12	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	12	0
0	0	0	0	0	12

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	10	0
0	0	0	0	8	4

12	0	0	0	0	0
3	1	0	0	0	0
0	0	12	0	0	0
0	0	0	12	0	0
0	0	0	0	9	3
0	0	0	0	3	4

8	1	0	0	0	0
2	5	0	0	0	0
0	0	7	12	0	0
0	0	11	6	0	0
0	0	0	0	9	3
0	0	0	0	8	4

12	8	0	0	0	0
3	1	0	0	0	0
0	0	8	6	0	0
0	0	9	5	0	0
0	0	0	0	7	11
0	0	0	0	11	6

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,8,0,0,0,0,0,4],[12,3,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,9,3,0,0,0,0,3,4],[8,2,0,0,0,0,1,5,0,0,0,0,0,0,7,11,0,0,0,0,12,6,0,0,0,0,0,0,9,8,0,0,0,0,3,4],[12,3,0,0,0,0,8,1,0,0,0,0,0,0,8,9,0,0,0,0,6,5,0,0,0,0,0,0,7,11,0,0,0,0,11,6] >;

(C₂×Dic₃).₉D₄ in GAP, Magma, Sage, TeX

(C_2\times {\rm Dic}_3)._9D_4

% in TeX

G:=Group("(C2xDic3).9D4");

// GroupNames label

G:=SmallGroup(192,217);

// by ID

G=gap.SmallGroup(192,217);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,64,590,387,100,6278]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^6=d^4=1,c^2=b^3,e^2=a*b^3,a*b=b*a,d*c*d^-1=a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=d*b*d^-1=e*b*e^-1=b^-1,e*c*e^-1=a*b^3*c,e*d*e^-1=b^3*d^-1>;

// generators/relations

G = (C2×Dic3).9D4 order 192 = 26·3

3rd non-split extension by C2×Dic3 of D4 acting via D4/C2=C22

G = (C₂×Dic₃).₉D₄ order 192 = 2⁶·3

3^rd non-split extension by C₂×Dic₃ of D₄ acting via D₄/C₂=C₂²